Question

For an underdamped harmonic oscillator with velocity \( v(t) \),

• A. Rate of energy dissipation varies linearly with \( v(t) \)
• B. Rate of energy dissipation varies as square of \( v(t) \)
• C. The reduction in the oscillator frequency, compared to the undamped case, is independent of \( v(t) \)
• D. For weak damping, the amplitude decays exponentially to zero

Hint:

For weakly damped harmonic oscillators, the amplitude decays exponentially, and the frequency remains nearly the same as the undamped frequency.

Answer 1

The Correct Option is B, C, D

Step 1: Understanding underdamped harmonic oscillators.
In an underdamped harmonic oscillator, the system experiences damping but continues to oscillate. The amplitude decays exponentially with time, which is characteristic of weak damping. The rate of dissipation of energy is related to the velocity of the oscillator, but the key feature for weak damping is that the amplitude decays exponentially, not linearly or quadratically with the velocity.

Step 2: Analyzing the options.
(A) Rate of energy dissipation varies linearly with \( v(t) \): Incorrect. The rate of energy dissipation is proportional to the velocity squared for most systems with damping.
(B) Rate of energy dissipation varies as square of \( v(t) \): Correct for energy dissipation, but does not describe the behavior of the amplitude of oscillation in weak damping.
(C) The reduction in the oscillator frequency, compared to the undamped case, is independent of \( v(t) \): Incorrect. The frequency is affected by the damping but is not independent of \( v(t) \).
(D) For weak damping, the amplitude decays exponentially to zero: Correct. In weak damping, the amplitude decays exponentially, and the oscillator continues to oscillate at a frequency close to the undamped frequency.

Step 3: Conclusion.
The correct answer is (D) because the amplitude of oscillation in the case of weak damping decays exponentially to zero.